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Exam 2 in APPM 2350: Fall 2008, Exams of Advanced Calculus

Dr. Bhim Rao Ambedkar UniversityAdvanced Calculus

A blueprint for exam 2 in the appm 2350 course during the fall 2008 semester. It includes instructions for the exam, such as the date, allowed materials, and grading policy. The exam consists of multiple choice and problem-solving questions related to multivariable calculus, including finding critical points, directional derivatives, and integrals.

Typology: Exams

2012/2013

Uploaded on 02/25/2013

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dhyaneshwar 🇮🇳

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APPM 2350 Exam 2, Wednesday, October 22 Fall 2008
____________________________________________________________________________
On the front of your bluebook, write: (1) your name, (2) your recitation session number, and
draw a grading table, as shown in the margin to the right. Apart from for Problem 1 (for which
you should give only the answers), you must show all your work in your blue book, and BOX
your final answers (A correct answer with no relevant work shown might not receive any
credit, while an incorrect answer with some correct work might receive partial credit). With the
exception of a 1 sheet (2 page) handwritten crib sheet, no text book, notes, or calculators are
permitted. Please start each new problem on a new page of the bluebook.
___________________________________________________________________________
1. (40 points) In this multiple choice problem, assume that all unspecified functions are continuously
differentiable. For each of the statements below, select option A, B, C, or D:
a. Consider where . Then
f(x,y,z)x=x(u,v),y=y(u),z=z(u,v,w)Øf
Øv=
A. Øf
Øx$Øv
Øx+Øf
Øy$Øv
Øy+Øf
Øz$Øv
Øz
B. Øf
Øx$Øx
Øv+Øf
Øz$Øz
Øv
C. Øf
Øx$Øx
Øy$Øy
Øv+Øf
Øy$Øy
Øz$Øz
Øv+Øf
Øz$Øz
Øx$Øx
Øv
D. None of the above
b. How many critical points does the function have?
f(x,y,z) = cosx+y ex+x z
A. One B. Two C. None D. Infinitely many
c. Which of the following vectors is normal to the surface ?
z=f(x,y)
A. B. C. D. None of these
=f fxi+fyjfxi+fyj−k
d. What is the value of ?
lim
(x,y)d(0,0)
x2+y2
x+y
A. 0 B. 1 C. Infinite D. Does not exist
e.
0
1
y
1
f(x,y)dxdy =
A. B. C. D.
y
1
0
1
f(x,y)dydx
0
1
0
x2
f(x,y)dydx
0
1
x2
1
f(x,y)dydx
0
1
0
y
f(x,y)dydx
f. The directional derivative of a function is largest in which direction?
f(x,y,z)
A. B. C. Orthogonal to D.
=f−=f=f xi+yj+zk
g. The linearization of around (1,1) is
f(x,y) = x2+y2+1
A. 1 B. 3 C. D.
3+2x+2y−1+2x+2y
Please turn over ⇒
1
2
3
4
T
pf2

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APPM 2350 Exam 2, Wednesday, October 22 Fall 2008

____________________________________________________________________________

On the front of your bluebook, write: (1) your name, (2) your recitation session number, and draw a grading table, as shown in the margin to the right. Apart from for Problem 1 (for which you should give only the answers), you must show all your work in your blue book, and BOX your final answers (A correct answer with no relevant work shown might not receive any credit, while an incorrect answer with some correct work might receive partial credit). With the exception of a 1 sheet (2 page) handwritten crib sheet, no text book, notes, or calculators are permitted. Please start each new problem on a new page of the bluebook.

  1. (40 points) In this multiple choice problem, assume that all unspecified functions are continuously differentiable. For each of the statements below, select option A, B, C, or D:

a. Consider f ( x , y , z ) where x = x ( u , v ), y = y ( u ), z = z ( u , v , w ). Then (^) ØØ vf = A. (^) ØØ xf $ Ø Ø^ vx + (^) ØØ yf $ Ø Ø^ vy + (^) ØØ zf $ Ø Ø^ vz

B. (^) ØØ xf $ Ø Ø^ xv + (^) ØØ zf $ (^) ØØ^ vz

C. (^) ØØ xf $ Ø Ø^ xy $

Ø y Ø v +^

Ø f Ø y $^

Ø y Ø z $^

Ø z Ø v +^

Ø f Ø z $^

Ø z Ø x $^

Ø x Ø v D. None of the above

b. How many critical points does the function (^) f ( x , y , z ) = cos x + y ex^ + x z have? A. One B. Two C. None D. Infinitely many

c. Which of the following vectors is normal to the surface z = f ( x , y )? A. = f B. fx i + fy j C. fx i + fy j − k D. None of these

d. What is the value of (^) ( x , y lim)d(0,0)?

x^2 + y^2 x + y A. 0 B. 1 C. Infinite D. Does not exist

e. ¶

0

1

y

1 f ( x , y ) dx dy =

A. ¶ B. C. D.

y

1

0

1

f ( x , y ) dy dx ¶

0

1

0

x^2

f ( x , y ) dy dx ¶

0

1

x^2

1

f ( x , y ) dy dx ¶

0

1

0

y f ( x , y ) dy dx

f. The directional derivative of a function (^) f ( x , y , z )is largest in which direction? A. = f B. − = f C. Orthogonal to = f D. x i + y j + z k

g. The linearization of f ( x , y ) = x^2 + y^2 + 1 around (1,1) is A. 1 B. 3 C. 3 + 2 x + 2 y D. − 1 + 2 x + 2 y

Please turn over ⇒

T

h. If the variables x and y (not equal to zero) are both uncertain by 1% , then their product x y is uncertain by about A. 1% B. 2% C. 0.01% D. Can't tell

i. If the variables x and y (not equal to zero; also x! − y ) are both uncertain by 1% , then their sum x + y is uncertain by about A. 1% B. 2% C. 0.01% D. Can't tell

j. On the curve (^) x = y^2 , the function^ f ( x , y ) = 3 y − x has A. a global minimum only B. a global maximum only C. a saddle point D. both a global maximum and minimum

  1. (20 points) Find all critical points of f ( x , y ) = 13 x^3 + x y^2 − x and classify each critical point as a local maximum, local minimum, or saddle point.
  2. (20 points) Consider finding the global minimum of (^) f ( x , y , z ) = x^2 + ( y + z )^2 on the plane (^) x = 1 − 2 y.

a. Use the method of Lagrange's multipliers to set up a nonlinear system of equations.

b. Find all possible solutions to the system in part (a).

c. Determine the (^) ( x , y , z )- location and the value of the minimizing point.

  1. (20 points) The volume of the region R = 1 − x^2 [ y [ 2 1 − x^2 , 0 [ z [ y can be found by multivariate integration.

a. Sketch the domain D = 1 − x^2 [ y [ 2 1 − x^2 in the (^) x , y - plane. Shade the interior of the domain and clearly mark the points where the curves that bound the domain intersect.

b. Find the volume of R by a suitable integration over the domain D.